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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2507.17084 |
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| _version_ | 1866918381073989632 |
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| author | Bickle, Allan Campbell, Russell |
| author_facet | Bickle, Allan Campbell, Russell |
| contents | In 1978, Anderson and White asked whether there is a decomposition of $K_{12}$ into two graphs, one planar and one toroidal. Using theoretical arguments and a computer search of all maximal planar graphs of order 12, we show that no such decomposition exists. We further show that if $G$ is planar of order 12 and $H\subseteq\overline{G}$ is toroidal, then $H$ has at least two fewer edges than $\overline{G}$. A computer search found all 123 unique pairs $\left(G,H\right)$ that make this an equality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17084 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Planar-Toroidal Decomposition of $K_{12}$ Bickle, Allan Campbell, Russell Combinatorics 05C10 In 1978, Anderson and White asked whether there is a decomposition of $K_{12}$ into two graphs, one planar and one toroidal. Using theoretical arguments and a computer search of all maximal planar graphs of order 12, we show that no such decomposition exists. We further show that if $G$ is planar of order 12 and $H\subseteq\overline{G}$ is toroidal, then $H$ has at least two fewer edges than $\overline{G}$. A computer search found all 123 unique pairs $\left(G,H\right)$ that make this an equality. |
| title | Planar-Toroidal Decomposition of $K_{12}$ |
| topic | Combinatorics 05C10 |
| url | https://arxiv.org/abs/2507.17084 |